t-test

When is the t-test used?

To test a claim about a population mean, we compare the sample mean with the claimed value. If the population standard deviation σ is unknown and the sample is small (usually n < 30), the standardised mean does not follow the normal distribution; it follows Student's t-distribution. The t-distribution is symmetric and bell-shaped like the normal curve but has heavier tails; it depends on the degrees of freedom and approaches the normal curve as these increase.

One-sample t-test

Assumption: the sample is random and drawn from an (approximately) normal population.

The test statistic is t = (x̄ − μ₀)/(s/√n), where x̄ is the sample mean, μ₀ is the hypothesised mean, n is the sample size and s is the sample standard deviation computed with divisor n − 1: s = √[Σ(x − x̄)²/(n − 1)]. The degrees of freedom are ν = n − 1.

Steps

1. State H₀: μ = μ₀ and H₁ (μ ≠ μ₀ for a two-tailed test; μ > μ₀ or μ < μ₀ for one-tailed). 2. Choose the level of significance α (say 5%). 3. Compute t. 4. Find the critical (table) value of t for n − 1 degrees of freedom. 5. If |t| > critical value, reject H₀; otherwise do not reject H₀ (the difference is not significant).

Example 1

n = 16, x̄ = 52, s = 4, H₀: μ = 50. t = (52 − 50)/(4/4) = 2 with 15 degrees of freedom.

Example 2

n = 10, x̄ = 48, s = 3, H₀: μ = 50, two-tailed at 5%. t = −2/(3/√10) ≈ −2.11. The table value for 9 degrees of freedom is 2.262. Since |t| < 2.262, H₀ is not rejected.

Key points

• Use t when σ is unknown and n is small.
• t = (x̄ − μ₀)/(s/√n), degrees of freedom n − 1.
• Compare |t| with the table value.
• "Not rejecting" H₀ does not prove H₀ true.

Exam tips

• Check whether s was computed with n or n − 1; use the n − 1 version in the formula above.
• Note whether the test is one-tailed or two-tailed before reading the table.